\2 1 Ai:Al \M OF COUPLES. 



i!. T,> jim/ th< rtsultant vf tiro couples not 



. 



Let the planes of the eouplcs intersect in the straight lino 



\vhioh is perpendicular to the plane of i r, and let 



the couples be referred to the common arm AB 9 and let their 

 forces tfi us altered be P and Q. 



In the plane of the paper draw Aa, Al at ri.^ht angles to 

 the planes of the couples P, Pand Q, Q; and equal in 1 

 to their axes. 



Let R be the resultant of the forces P and Q at A, acting 

 in the direction AR\ and of P and Q at /A ueting in the 

 direction #72. 



Since ^4P, .4$ are parallel to BP, BQ respectively, t; 

 fore AR is parallel to 111!. 



Hence the two couples are equivalent to the single couple 

 It, R acting on the arm AB. 



Draw Ac perpendicular to the plane of 7?, 7?, and in the 

 same proportion to Aa, AI> that the moment of the < ,, ij.le 

 // A' is to those of P, P and Q, Q respectively. Thru Ac 



M "i //. y/. Now the three straight linos A>- 

 the same angles with each other that Al\ Al>. 

 make with each other; also they are in <>por- 



tion in whidi AJB.P, AB . It. All. ',> ;uv; that is in which 

 P t A', Q are. 



R is the resultant of P and (J- therefore Ac is the 

 diagonal of the parallelogram on Aa, Ab (see Art. 17). 



li.-nco if two straight lines, having a common extremity. 

 represent the axes of two couplos. that diagonal of the paral- 

 lelogram described on these straight lines as adjacent Md--s 

 which passes through their common extremity represents the 

 axis of the resultant couple. 



